Hey Martin,
Gladly, but I can’t always answer promptly. But in general I always reply, especially to people who have previously used the search function to benefit from what is already written 
It wouldn’t work this way. If you look at the specification of the relay outputs AL{0..2} in the VFD manual, then you can determine (already without having to open the VFD) what kind of relay is behind these connectors:
As you can see there is a minimum load current and that is a sure sign that it is a solid state relay. Unlike electromagnetic relays, a solid-state relay can only hold its state when a minimal load current is flowing. That’s because they’re not really switches, they’re thyristors (a specific type of semiconductor). With electromagnetic relays however, there is no requirement for a minimum amount of load current, because they are really just a switch, i.e. mechanical contacts.
You can also see from the further technical data of the AL{0..2} connectors that these are outputs for higher voltage and current loads:
Terminal Name Description Ratings AL0
AL1
AL2Relay common contact
Relay contact, normally open
Relay contact, normally closed250VAC, 2.5A (R load) max.
250VAC, 0.2A (I load, P.F.=0.4) max.
100VAC, 10mA min.
30VDC, 3.0A (R load) max.
30VDC, 0.7A (I load, P.F.=0.4) max.
5VDC, 100mA min.
You should always first consider what you actually want to switch here: You want to trigger “Estop” (pin 23) on the DB-25 I/O port of the Onefinity controller, which means you want to set an input of the AVR microcontroller to logical high or low directly which practically means connect pin 23 (estop) to pin 25 (ground). Only an extremely small amount of current flows for this, which means that a solid-state relay is ruled out and the electromagnetic relay that you would use for this only needs to be as small as possible. Also with an electromagnetic relay, the switched contacts are galvanically isolated from control coil (even without using an optocoupler) which means you don’t have to worry about ground.
As you may have already read in the manual, you can assign all output functions, including a logical combination of several functions (thanks to the built-in logic unit available inside the VFD) that you can assign to the AL{0..2} outputs, to the other outputs as well, these are terminals 11 and 12 on the Hitachi WJ200 aka Omron MX2 VFDs. These are open collector outputs (see also schematic below). This means that here you can do without a driver transistor for controlling the control coil of the relay, because the transistor is practically already built-in inside the VFD.
The manual also states explicitly for what to use the different output terminals:
Use an open collector output (terminal [11] or [12]) for a low-current logic
signal interface or to energize a small relay (50 mA maximum). Use the relay output to interface to higher voltage and current devices (10 mA minimum).– Source: Hitachi WJ200 Series Inverter Instruction Manual
You can however use the outputs 11 and 12 for relays as strong as you want, als long as you provide an appropriate driver circuit in front of the control coil.
A suitable relay for your application would be, for example, the Omron G5V-1 Ultra-miniature, Highly Sensitive SPDT Relay for Signal Circuits (datasheet):
As you can see, it is very, very small! The coin in the background is approx. one inch in diameter (25.75 mm to be exact). Shown is the 12 V version but here I would use the 24 V version. However this is a single electronics component, made for being soldered into a circuit board. I don’t know if you are able to do this, you could have someone who can do this for you, but there also could be other solutions e.g. buying a DIN-rail 24 V relay. In this case there is a driver transistor in front of the relay coil inside the relay module.
Because of open collector output on the VFD, the wiring of the bare relay is therefore very simple, as shown here on page 4-51:
You can use either of the outputs, 11 or 12. Apart from the relay you only need a voltage source (a separate power supply) and a flyback diode.
Note that output 11 on Hitachi WJ200 aka Omron MX2 can also be used as part of EDM (external device monitoring) if you want to comply to ISO 13849-1 and want to use a safety relay like Omron G9SE. See Appendix E-2 of your Hitachi VFD Manual or the safety wiring shown here.
Basically, if you want to operate relays, for example even fatter ones later, for example to switch your spindle water cooling or the dust extraction system automatically on and off, you should buy a power supply for relays right away.
A 24 V switching power supply for DIN Rail is usual here. You don’t use 5 V or 12 V because the higher the voltage, the lower the currents (with the same power), so 24 V is usual here. The WJ200/MX2 also has an internal 24 V power source, but its power is very limited and not designed for driving higher loads.
Note: Attention! Operating a relay here on an output of the VFD does not require any additional components like a driver because this is an open collector output. However if you want to drive a bare relay from one of the pins configured as outputs on the DB-25 I/O port of the Onefinity controller, which mostly are direct in- and outputs of the AVR microcontroller and that are not open-collector outputs, a transistor driver circuit is required!
Shielded and twisted, you can’t do it any better.
This serial connection is a balanced line, which means it already should work relatively well without twisting and shielding, but with it it’s perfect.
You connect the positive terminal of the VFD (SP) to the positive pin (pin 13) of DB-25 I/O port of the Onefinity Controller and the negative terminal (SN) to the negative pin on the other end (pin 14).